The concert flute or orchestral Boehm flute used by flautists has a conical headjoint and a cylindrical body. With the current state of technology the piccolo, which sounds an octave higher and also belongs to the family of orchestral Boehm flutes, uses one of the following two designs:                1. For classical music—including symphony and chamber orchestras—it always has a conical body and a cylindrical headjoint (the so-called conical Boehm piccolo), which extends to low d (d1). This is currently the most common design.        Older designs, for example German Patent No. DE 3502842, have two thumb holes, and more recent designs one thumb hole which fixes the shortcomings associated with the two thumb holes concerning acoustic, phonetic and speech in the third octave (register). There have been experimental designs extending to low c (c1) and with a double c2 (sounding c3) thumb hole, but they have featured a poor response and intonation, and unevenness between the registers. They have been rejected by musicians and have not become established.        Or        2. For marching music and brass bands/orchestras the piccolo has a cylindrical body and a conical headjoint (the so-called cylindrical Boehm piccolo). This model uses one of two designs: firstly with two thumb holes and extending to low d (d1) and low c (c1). Secondly with one thumb hole and extending to low c (c1), as described in German-Patent No. DE 19810520.        
The explanation for the different main uses is to be found in human perception of sound.
The timbre is determined by the overtone structure, i.e. its composition and the strengths of the different overtones, and is affected by the geometrical shape of the tube. There is thus a fundamental difference in timbre and overtone structure between so-called cylindrical flutes and so-called conical flutes. Whilst cylindrical instruments can produce the entire overtone series more or less strongly and without any gaps, the overtone structure of conical instruments is somewhat restricted to the odd-numbered partials. The high register of conical instruments thus sounds more pleasant to the human ear and is therefore preferred for classical music.
The conical piccolo flutes differ from the cylindrical piccolo flutes in terms of acoustic. By the conical narrowing the piccolos become a little deeper and must therefore be, compared to the cylindrical piccolo flutes, made somewhat shorter. Accordingly the tone holes must be placed a little higher to the embouchure. The way to find the exact measurements of the shortening or the distances between the embouchure and tone holes, is the empirical or experimentally. Speculative calculations would be too vague here, because it is important to note that the tone holes serve not only as tone holes but also have sound-wave node-building functions especially in the third, and partially also in the second register. There must be found a compromise for tone, response and intonation. This is only possible through experiments and by trying.
Since the establishment of conical Boehm piccolos in the orchestral, 150 years ago, musicians have wanted a conical Boehm piccolo extending to low c (c1) (sounding c2). Thus nearly all prominent flute makers have worked on the problem, though without succeeding in developing an instrument that meets the timbre requirements for an orchestra. These instruments had, by deepening to deep c1, in the first or low register difficulties with the sound and in the third octave problems with the intonation and especially with the response. This explains why piccolos were built not deeper than d1.
Compositions by Benjamin Britten, Engelbert Humperdinck, Gustav Mahler, Ottorino Respighi, Arnold Schoenberg and Giuseppe Verdi call for a piccolo range extending to c1. Particularly in contemporary music the piccolo is assuming an ever more important role as a solo instrument, e.g. in pieces by Franco Donatoni, Brian Ferneyhough, Jan Huylebroeck, Lowell Liebermann and Karlheinz Stockhausen.
However, without changes to the original compositions the current state of technology does not permit performance of such pieces utilising the timbre of the conical piccolo.
Science has not hitherto been in a position to fully explain the acoustic conditions of vibrating air columns enclosed in instrument tubes. It fails when it comes to giving the instrument maker absolute, reliably calculated figures. Thus constructing an instrument with good tuning and a beautiful sound remains a major problem that can only in part be resolved by using the calculated figures as a guideline, and can better be achieved through experiment and by use of empirically gained knowledge.
The totally different considerations regarding conical piccolos extending to low d (d1) and conical piccolos extending to low c (c1) stem from the fact that most of the parameters regarding woodwind instruments and in particular piccolos can not yet been explained. This is in part attributable to the fact that the flautist's anatomy is to a large extent integrated into and involved in creating the notes. The shaping of the lips, the embouchure and many other factors all play a role, and these considerations cannot easily be registered physically and mathematically. The factors and functions affecting the calculation are major and complex, and their influence has not yet been precisely grasped, thus no usable mathematical solutions are known to date. The conical piccolo going down to c1 does not just represent an extension of the model going down to d1, but is an instrument whose entire construction must be developed using a completely new hole-setting scheme.
On the current state there are no scientific findings on which to base a method of calculation for establishing a hole-setting scheme for conical piccolos. The hole-setting scheme represents more than a known physico-mathematical relationship. There are no concrete solutions to be found in the literature. Proposals alone, as stated in the following literature, are not yet the solution to the problem. A piccolo constructed in accordance with such a proposal could not be used as a concert piccolo, and would at most be feasible as a mass-produced article, e.g. for educational purposes.
The known methods of calculation for establishing a tone-hole arrangement for a wind instrument or flute are laid down in scientific treatises, viz Otto Steinkopfs ‘Zur Akustik der Blasinstrumente’ [On the acoustics of wind instruments] (Celle 1983), C. N. Nederveen's ‘Acoustical aspects of woodwind instruments’ (Amsterdam 1969) and A. H. Benade's ‘On Woodwind Instrument Bores in Journal of the Acoustical Society of America’, Vol. 31, No. 2, 18.11.1957. As regards the above statements, Steinkopf states on p 26 of the above book in the chapter ‘The arrangement of the tone holes (Boehm flute)’: “Totally reliable figures, however, are hardly to be expected in this context, thus one cannot entirely dispense with empirically gained knowledge.”
The evidence for this can also be seen in the fact that the global market has not yet witnessed a usable conical piccolo extending to low c (c1) and with only one c2 hole. Instrument makers have long been seeking a resolution to this problem, and this invention forms the basis for such a resolution.
A further problem is that despite the fact that in 1939 international concert pitch was set at a1=440 Hz in London, it has not been uniformly defined for all countries, thus different piccolos are still being built.